In the present paper, we demonstrate 3-variable 2-parameter $q$-Hermite polynomials via generating functions along with their series definitions, $q$-derivatives, operational identities, then we deduce some properties for 2-variable 1-parameter $q$-Hermite polynomials. Also, we present the same mentioned features for multi-index $q$-Hermite polynomials and their associated formalism. Moreover, we utilize the techniques of quasi-monomial extension to explain and implement $q$-multiplicative and $q$-derivative operators for $q$-Hermite polynomials in three variables and multi-index $q$-Hermite polynomials. Finally, we present applications that can be derived using these polynomials, where the graphs of the zero functions and the meshes are displayed.

Certain properties of generalized and Higher–order q-Hermite polynomials: monomiality and applications to their zero distributions

William Ramirez
;
Clemente Cesarano;Pablo Buitron
2025-01-01

Abstract

In the present paper, we demonstrate 3-variable 2-parameter $q$-Hermite polynomials via generating functions along with their series definitions, $q$-derivatives, operational identities, then we deduce some properties for 2-variable 1-parameter $q$-Hermite polynomials. Also, we present the same mentioned features for multi-index $q$-Hermite polynomials and their associated formalism. Moreover, we utilize the techniques of quasi-monomial extension to explain and implement $q$-multiplicative and $q$-derivative operators for $q$-Hermite polynomials in three variables and multi-index $q$-Hermite polynomials. Finally, we present applications that can be derived using these polynomials, where the graphs of the zero functions and the meshes are displayed.
2025
Quantum calculus, Extending the monomiality principle, q-Hermite polynomials, 3- variable 2-parameter Hermite polynomials, Higher order Hermite polynomials, q-dilatation operator.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.14086/8561
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